In this section, we first present two cases based on the viscous Burgers' equation: \(u_t= \nu u_{xx} -uu_x\) . The first case consists in generating three spatiotemporal datasets with an identical initial condition (Dirac delta) and different viscosity parameters \(\nu = \lbrace 0.1,0.2,0.4\rbrace \) , see figure REF (a). For the second case we fix \(\nu =1\)  and vary the initial conditions (Dirac delta, periodic and step-like functions) see figure REF (b). These datasets are generated from analytical solutions [1]}, on top of which we add a high noise level of \(10 \%\)  Gaussian white noise. In both cases, 40 points in time and 50 points in time are considered leading to 2000 samples per experiment. In such conditions, the observations of the fields \(u\)  are coarsely sampled and noisy enough such that it would not be possible to compute accurate enough derivatives for the libraries using classical methods such as finite differences or polynomial interpolation. By leveraging automatic differentiation and physics informed neural networks to compute the libraries, these cases are typically the ones where deep learning based model discovery frameworks excel. We compare the two different methods for promoting sparsity in DeepMod described in the previous sections: the individual sparsity promotion method is performed \(q\)  times (once per experiment) or the grouped sparsity promotion is performed, see figures REF  and REF . The sparsity patterns that are recovered might be different, see figure REF (a): the individual randomised adaptive Lasso is incapable of recovering the nonlinear advection term for the last experiment. This can be explained physically, as it happens for the experiment with the highest diffusion coefficient (\(\nu =0.4\) ): diffusion is predominant over nonlinear advection. On the other hand, by promoting grouped sparsity, the ground truth underlying PDE can be recovered and it can be verified that it actually generalises better since \(\text{MSE}_{\text{test,group}} = 1.1749e^{-3} < \text{MSE}_{\text{test,individual}} = 1.1761e^{-3}\) . We report a similar conclusion for the chaotic Kuramoto-Sivashinsky equation, \(\text{MSE}_{\text{test,group}} = 5.51e^{-3} < \text{MSE}_{\text{test,individual}} = 5.59e^{-3}\) , see figure REF . The same hyperparameters were used, see Appendix  and 2 \(\times \)  2000 points are randomly sampled from a numerical solution [2]}, on top of which we add a high noise level of \(20 \%\)  Gaussian white noise. Since we use NNs to interpolate across samples, we also investigated the impact of sharing NN layers across tasks for cases 1 and 2. Interestingly, sharing layers across neural networks does not help DeepMod finding better coefficient estimates nor improves the variable selection performance - it can actually be the opposite, see figure REF  and Appendix .
